A130263 Number of degree-n permutations such that number of cycles of size k is odd (or zero) for every k.

1, 1, 1, 6, 14, 85, 529, 3451, 26816, 243909, 2507333, 26196841, 323194816, 4086482335, 57669014597, 864137455455, 13792308331616, 231648908415001, 4211676768746569, 79205041816808905, 1584565388341689032, 33265011234209710011, 730971789582886971689

Offset: 0

Author

Vladeta Jovovic, Aug 06 2007

Keywords

• easy
• nonn

Examples

			a(2)=1 because we have (12) ((1)(2) does not qualify). a(4)=14 because the following 10 permutations of 4 do not qualify: (1)(2)(3)(4), (14)(2)(3), (1)(24)(3), (1)(2)(34), (13)(2)(4), (13)(24), (1)(23)(4), (14)(23), (12)(3)(4) and (12)(34).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A055922, A130219, A130220, A130268, A218504, A270597, A270598, A270614.

Programs

Magma

m:=40;
f:= func< x | (&*[1 + Sinh(x^j/j): j in [1..m+1]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Mar 18 2023

Maple

g:=product(1+sinh(x^k/k),k=1..40): gser:=series(g,x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..21); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or irem(j, 2)=1, multinomial(n, n-i*j, i$j)
       *(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

Mathematica

nn = 25; Range[0, nn]!*CoefficientList[Series[Product[1 + Sinh[x^k/k], {k, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 20 2016 *)

SageMath

m=40
def f(x): return product( 1 + sinh(x^j/j) for j in range(1,m+2) )
def A130263_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A130263_list(m) # G. C. Greubel, Mar 18 2023

Formula

E.g.f.: Product_{k>0} (1+sinh(x^k/k)).

a(n) ~ c * n!, where c = A270614 = Product_{k>=1} ((1 + sinh(1/k)) / exp(1/k)) = 0.625635801977949844... . - Vaclav Kotesovec, Mar 20 2016

Extensions

More terms from Emeric Deutsch, Aug 24 2007